# American Institute of Mathematical Sciences

March  2020, 9(1): 1-25. doi: 10.3934/eect.2020014

## Exact boundary observability and controllability of the wave equation in an interval with two moving endpoints

 Laboratory of Functional Analysis and Geometry of Spaces, Faculty of Mathematics and Computer Sciences, University of M'sila, 28000, Algeria

Received  November 2017 Revised  May 2018 Published  October 2019

Fund Project: The author is supported by the Algerian Ministry of Higher Education and Scientific Research, under a PRFU Project No. C00L03UN280120180007.

We study the wave equation in an interval with two linearly moving endpoints. We give the exact solution by a series formula, then we show that the energy of the solution decays at the rate $1/t$. We also establish observability results, at one or at both endpoints, in a sharp time. Moreover, using the Hilbert uniqueness method, we derive exact boundary controllability results.

Citation: Abdelmouhcene Sengouga. Exact boundary observability and controllability of the wave equation in an interval with two moving endpoints. Evolution Equations & Control Theory, 2020, 9 (1) : 1-25. doi: 10.3934/eect.2020014
##### References:
 [1] N. L. Balazs, On the solution of the wave equation with moving boundaries, J. Math. Anal. Appl., 3 (1961), 472-484.  doi: 10.1016/0022-247X(61)90071-3.  Google Scholar [2] C. Bardos and G. Chen, Control and stabilization for the wave equation. Ⅲ: Domain with moving boundary, SIAM J. Control Optim., 19 (1981), 123-138.  doi: 10.1137/0319010.  Google Scholar [3] G. Birkhoff and G.-C. Rota, Ordinary Differential Equations, 4$^{th}$ edition, John Wiley & Sons, Inc., New York, 1989.  Google Scholar [4] J. Cooper and C. Bardos, A nonlinear wave equation in a time dependent domain, J. Math. Anal. Appl., 42 (1973), 29-60.  doi: 10.1016/0022-247X(73)90120-0.  Google Scholar [5] L. Z. Cui, X. Liu and H. Gao, Exact controllability for a one-dimensional wave equation in non-cylindrical domains, J. Math. Anal. Appl., 402 (2013), 612-625.  doi: 10.1016/j.jmaa.2013.01.062.  Google Scholar [6] V. V. Dodonov, A. B. Klimov and D. E. Nikonov, Quantum phenomena in resonators with moving walls, J. Math. Phys., 34 (1993), 2742-2756.  doi: 10.1063/1.530093.  Google Scholar [7] E. Knobloch and R. Krechetnikov, Problems on time-varying domains: Formulation, dynamics, and challenges, Acta Appl. Math., 137 (2015), 123-157.  doi: 10.1007/s10440-014-9993-x.  Google Scholar [8] V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, RAM: Research in Applied Mathematics, Masson, Paris, John Wiley & Sons, Ltd., Chichester, 1994.  Google Scholar [9] V. Komornik and P. Loreti, Fourier Series in Control Theory, Springer Monographs in Mathematics, Springer-Verlag, New York, 2005. doi: 10.1007/b139040.  Google Scholar [10] J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod-Gautier Villars, Paris, 1969.  Google Scholar [11] J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systemes Distribués. Tome 1. Contrôlabilité Exacte, Recherches en Mathématiques Appliquées, 8. Masson, Paris, 1988.  Google Scholar [12] L. Q. Lu, S. J. Li, G. Chen and P. F. Yao, Control and stabilization for the wave equation with variable coefficients in domains with moving boundary, Systems & Control Lett., 80 (2015), 30-41.  doi: 10.1016/j.sysconle.2015.04.003.  Google Scholar [13] L. A. Medeiros, J. Limaco and S. B. Menezes, Vibrations of elastic strings: Mathematical aspects, part two, J. Comput. Anal. Appl., 4 (2002), 211-263.  doi: 10.1023/A:1013151525487.  Google Scholar [14] M. Milla Miranda, Exact controllability for the wave equation in domains with variable boundary, Rev. Mat. Complut. Madrid, 9 (1996), 435-457.  doi: 10.5209/rev_rema.1996.v9.n2.17595.  Google Scholar [15] G. T. Moore, Quantum theory of electromagnetic field in a variable-length one-dimensional cavity, J. Math. Phys., 11 (1970), 2679-2691.  doi: 10.1063/1.1665432.  Google Scholar [16] M. A. Pinsky, Introduction to Fourier Analysis and Wavelets, Graduate Studies in Mathematics, 102. American Mathematical Society, Providence, RI, 2009. doi: 10.1090/gsm/102.  Google Scholar [17] D. L. Russell, Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions, SIAM Rev., 20 (1978), 639-739.  doi: 10.1137/1020095.  Google Scholar [18] A. Sengouga, Observability and controllability of the 1-D wave equation in domains with moving boundary, Acta Appli. Math., 157 (2018), 117-128.  doi: 10.1007/s10440-018-0166-1.  Google Scholar [19] A. Sengouga, Observability of the 1-D wave equation with mixed boundary conditions in a non-cylindrical domain, Mediterr. J. Math., 15 (2018), Art. 62, 22 pp. doi: 10.1007/s00009-018-1107-y.  Google Scholar [20] H. C. Sun, H. F. Li and L. Q. Lu, Exact controllability for a string equation in domains with moving boundary in one dimension, Electron. J. Diff. Equations, 2015 (2015), 7 pp.  Google Scholar [21] P. K. C. Wang, Stabilization and control of distributed systems with time-dependent spatial domains, J. Optim. Theory Appl., 65 (1990), 331-362.  doi: 10.1007/BF01102351.  Google Scholar [22] E. Zuazua, Propagation, observation, and control of waves approximated by finite difference methods, SIAM Rev., 47 (2005), 197-243.  doi: 10.1137/S0036144503432862.  Google Scholar

show all references

##### References:
 [1] N. L. Balazs, On the solution of the wave equation with moving boundaries, J. Math. Anal. Appl., 3 (1961), 472-484.  doi: 10.1016/0022-247X(61)90071-3.  Google Scholar [2] C. Bardos and G. Chen, Control and stabilization for the wave equation. Ⅲ: Domain with moving boundary, SIAM J. Control Optim., 19 (1981), 123-138.  doi: 10.1137/0319010.  Google Scholar [3] G. Birkhoff and G.-C. Rota, Ordinary Differential Equations, 4$^{th}$ edition, John Wiley & Sons, Inc., New York, 1989.  Google Scholar [4] J. Cooper and C. Bardos, A nonlinear wave equation in a time dependent domain, J. Math. Anal. Appl., 42 (1973), 29-60.  doi: 10.1016/0022-247X(73)90120-0.  Google Scholar [5] L. Z. Cui, X. Liu and H. Gao, Exact controllability for a one-dimensional wave equation in non-cylindrical domains, J. Math. Anal. Appl., 402 (2013), 612-625.  doi: 10.1016/j.jmaa.2013.01.062.  Google Scholar [6] V. V. Dodonov, A. B. Klimov and D. E. Nikonov, Quantum phenomena in resonators with moving walls, J. Math. Phys., 34 (1993), 2742-2756.  doi: 10.1063/1.530093.  Google Scholar [7] E. Knobloch and R. Krechetnikov, Problems on time-varying domains: Formulation, dynamics, and challenges, Acta Appl. Math., 137 (2015), 123-157.  doi: 10.1007/s10440-014-9993-x.  Google Scholar [8] V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, RAM: Research in Applied Mathematics, Masson, Paris, John Wiley & Sons, Ltd., Chichester, 1994.  Google Scholar [9] V. Komornik and P. Loreti, Fourier Series in Control Theory, Springer Monographs in Mathematics, Springer-Verlag, New York, 2005. doi: 10.1007/b139040.  Google Scholar [10] J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod-Gautier Villars, Paris, 1969.  Google Scholar [11] J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systemes Distribués. Tome 1. Contrôlabilité Exacte, Recherches en Mathématiques Appliquées, 8. Masson, Paris, 1988.  Google Scholar [12] L. Q. Lu, S. J. Li, G. Chen and P. F. Yao, Control and stabilization for the wave equation with variable coefficients in domains with moving boundary, Systems & Control Lett., 80 (2015), 30-41.  doi: 10.1016/j.sysconle.2015.04.003.  Google Scholar [13] L. A. Medeiros, J. Limaco and S. B. Menezes, Vibrations of elastic strings: Mathematical aspects, part two, J. Comput. Anal. Appl., 4 (2002), 211-263.  doi: 10.1023/A:1013151525487.  Google Scholar [14] M. Milla Miranda, Exact controllability for the wave equation in domains with variable boundary, Rev. Mat. Complut. Madrid, 9 (1996), 435-457.  doi: 10.5209/rev_rema.1996.v9.n2.17595.  Google Scholar [15] G. T. Moore, Quantum theory of electromagnetic field in a variable-length one-dimensional cavity, J. Math. Phys., 11 (1970), 2679-2691.  doi: 10.1063/1.1665432.  Google Scholar [16] M. A. Pinsky, Introduction to Fourier Analysis and Wavelets, Graduate Studies in Mathematics, 102. American Mathematical Society, Providence, RI, 2009. doi: 10.1090/gsm/102.  Google Scholar [17] D. L. Russell, Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions, SIAM Rev., 20 (1978), 639-739.  doi: 10.1137/1020095.  Google Scholar [18] A. Sengouga, Observability and controllability of the 1-D wave equation in domains with moving boundary, Acta Appli. Math., 157 (2018), 117-128.  doi: 10.1007/s10440-018-0166-1.  Google Scholar [19] A. Sengouga, Observability of the 1-D wave equation with mixed boundary conditions in a non-cylindrical domain, Mediterr. J. Math., 15 (2018), Art. 62, 22 pp. doi: 10.1007/s00009-018-1107-y.  Google Scholar [20] H. C. Sun, H. F. Li and L. Q. Lu, Exact controllability for a string equation in domains with moving boundary in one dimension, Electron. J. Diff. Equations, 2015 (2015), 7 pp.  Google Scholar [21] P. K. C. Wang, Stabilization and control of distributed systems with time-dependent spatial domains, J. Optim. Theory Appl., 65 (1990), 331-362.  doi: 10.1007/BF01102351.  Google Scholar [22] E. Zuazua, Propagation, observation, and control of waves approximated by finite difference methods, SIAM Rev., 47 (2005), 197-243.  doi: 10.1137/S0036144503432862.  Google Scholar
Extension of an initial data $\phi ^{0}$ when $v _{1}<v _{2}$
Propagation of a wave with a small support near an endpoint $(v _{2}<v _{1})$
Propagation of small disturbances with supports near one or two ends $(v _{1}<v _{2})$
 [1] Teresa D'Aprile. Bubbling solutions for the Liouville equation around a quantized singularity in symmetric domains. Communications on Pure & Applied Analysis, 2021, 20 (1) : 159-191. doi: 10.3934/cpaa.2020262 [2] Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384 [3] Xinyu Mei, Yangmin Xiong, Chunyou Sun. Pullback attractor for a weakly damped wave equation with sup-cubic nonlinearity. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 569-600. doi: 10.3934/dcds.2020270 [4] Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243 [5] Yongxiu Shi, Haitao Wan. Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case. Electronic Research Archive, , () : -. doi: 10.3934/era.2020119 [6] Mostafa Mbekhta. Representation and approximation of the polar factor of an operator on a Hilbert space. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020463 [7] Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319 [8] Abdollah Borhanifar, Maria Alessandra Ragusa, Sohrab Valizadeh. High-order numerical method for two-dimensional Riesz space fractional advection-dispersion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020355 [9] Oussama Landoulsi. Construction of a solitary wave solution of the nonlinear focusing schrödinger equation outside a strictly convex obstacle in the $L^2$-supercritical case. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 701-746. doi: 10.3934/dcds.2020298 [10] Pierre-Etienne Druet. A theory of generalised solutions for ideal gas mixtures with Maxwell-Stefan diffusion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020458 [11] Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444 [12] Lingwei Ma, Zhenqiu Zhang. Monotonicity for fractional Laplacian systems in unbounded Lipschitz domains. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 537-552. doi: 10.3934/dcds.2020268 [13] Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020049 [14] Touria Karite, Ali Boutoulout. Global and regional constrained controllability for distributed parabolic linear systems: RHUM approach. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020055 [15] Mehdi Bastani, Davod Khojasteh Salkuyeh. On the GSOR iteration method for image restoration. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 27-43. doi: 10.3934/naco.2020013 [16] Soniya Singh, Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of second order impulsive systems with state-dependent delay in Banach spaces. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020103 [17] Lin Shi, Xuemin Wang, Dingshi Li. Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5367-5386. doi: 10.3934/cpaa.2020242 [18] Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217 [19] Adel M. Al-Mahdi, Mohammad M. Al-Gharabli, Salim A. Messaoudi. New general decay result for a system of viscoelastic wave equations with past history. Communications on Pure & Applied Analysis, 2021, 20 (1) : 389-404. doi: 10.3934/cpaa.2020273 [20] Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320

2019 Impact Factor: 0.953